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%I #39 May 06 2022 13:13:51
%S 138140141,180182183,240242243,250252253,330332333,400402403,
%T 408410411,478480481,546548549,570572573,600602603,646648649,
%U 660662663,676678679,768770771,838840841,876878879,928930931,940942943,970972973,996998999,100810101011,109610981099
%N Primes formed by the concatenation of exactly three successive composite numbers.
%C The primes that are obtained by the concatenation of exactly three successive composite numbers are always of the form c||c+2||c+3, with c+1 prime and c+3 odd <> 5, hence c must necessary ends with 0, 6, 8 (see examples).
%C No such primes can be obtained with the two other possible configurations of 3 successive composite numbers: c||c+1||c+2 or c||c+1||c+3.
%C The number of digits in each term is a multiple of 3. If a term existed for which this were not true, then c would necessarily be of the form 10^k - 2 (A099150), but then c+1 = 10^k - 1 would not be prime.
%H Jon E. Schoenfield, <a href="/A342214/b342214.txt">Table of n, a(n) for n = 1..10000</a>
%H G. L. Honaker, Jr. and Chris Caldwell, <a href="https://primes.utm.edu/curios/page.php/138140141.html">Prime Curios! 138140141</a>.
%e a(1) = 138140141 because 138, 140, 141 are 3 successive composite numbers, then concat(138, 140, 141) = 138140141 is prime and is the least prime with this property (see link Prime Curios!).
%e The smallest such primes whose first composite ends respectively with 0, 6, 8 are: a(2) = 180182183, a(9) = 546548549, a(1) = 138140141.
%e If (3,q) is the smallest term formed by the concatenation of 3 successive composite numbers with each q digits: (3,3) = a(1) = 138140141, (3,4) = a(22) = 100810101011.
%t nextc[n_] := Module[{k = n + 1}, While[PrimeQ[k], k++]; k]; seq = {}; n1 = 4; n2 = nextc[n1]; Do[n3 = nextc[n2]; c = FromDigits @ Flatten @ Join[IntegerDigits /@ {n1, n2, n3}]; If[PrimeQ[c], AppendTo[seq, c]]; n1 = n2; n2 = n3, {1000}]; seq (* _Amiram Eldar_, Mar 05 2021 *)
%o (PARI) lista(nn) = {my(ca=4, cb=6); forcomposite(c=7, nn, if (isprime(x=eval(concat(Str(ca), concat(Str(cb), Str(c))))), print1(x, ", ")); ca = cb; cb = c;);} \\ _Michel Marcus_, Mar 05 2021
%o (Python)
%o from sympy import isprime
%o def aupto(limit):
%o c, t, alst = 6, 689, []
%o while t < limit:
%o t = int("".join(map(str, [c, c+2, c+3])))
%o if isprime(c+1) and not isprime(c+3) and isprime(t): alst.append(t)
%o c += [6, 4, 2, 2, 2][(c%10)//2]
%o return alst
%o print(aupto(109610981099)) # _Michael S. Branicky_, Mar 05 2021
%Y Cf. A087341, A281684, A342049 (similar, with 2 consecutive composites).
%K nonn,base
%O 1,1
%A _Bernard Schott_, Mar 05 2021
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